Re: Attaction of Fixed Points
- From: "Ioannis" <morpheus@xxxxxxxxxxxx>
- Date: Fri, 10 Jun 2005 20:05:47 +0300
Ο "Alexander S Klein" <asklein@xxxxxxxxxxx> έγραψε στο μήνυμα
news:Pine.GSO.4.05.10506101227150.23648-100000@xxxxxxxxxxxxxxxxxxxxxxxxx
>Could you break this down a bit more? I'm still having a hard time
>understanding this. I still don't understand how I would figure out how
>to determine how many iterations it would take to get to the fixed point
>of systems with weak and strong attractions?
I am not sure if there is a non-empirical way to tell "how many" iterations
one would need for any particular system for convergence. One thing you can
tell with certainty, is that the closer to zero the modulus of the
multiplier of a fixed point is, the fastest the convergence.
For a fixed point x_0, the modulus of the multiplier is |f'(x_0)|, where f
is your system function (on a single iterative system) so if you have two
fixed points x_0 and x_1 for f, you can be sure that if for example 0 <
|f'(x_0)| < |f'(x_1)| < 1, then convergence at x_0 will require *less*
iterations in general, but you cannot know *how many* less, exactly.
The fastest convergence occurs when |f'(x)| = 0 for some fixed point x, and
in this case x is called a "super-attracting" fixed point. In this case, the
convergence is really fast, in the order of a couple of iterations.
No conclusion can be drawn for two fixed points x_0 and x_1, if the
multiplier of one is unity. I.e., if 0 < |f'(x_0)| < |f'(x_1)| = 1.
> Thanks in advance for any help.
> Alex.
--
I. N. Galidakis
http://users.forthnet.gr/ath/jgal/
Eventually, _everything_ is understandable
[snip]
.
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- Attaction of Fixed Points
- From: Alexander S Klein
- Attaction of Fixed Points
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